Volume 99A, number 1 PHYSICS LETTERS 14 November 1983

CLASSICAL PERIODIC SOLUTIONS OF THE EQUALMASS 2n-BODY PROBLEM,

b-ION PROBLEM AND THE n-ELECTRON ATOM PROBLEM

Ian DAVIES, Aubrey TRUMAN and David WILLIAMS Department of Mathematics, University College of Swansea, Si&eton Park, Swansea SA2 8PP, UK

Received 1.5 September 1983

We present a new family of classical periodic orbits for physically interesting hamiltonian systems, such as the Zeeman effect hamiltonian for the n-electron atom.

This letter gives an account of what we have dubbed “general&d Lagrange solutions” of the equal-mass

H(%P) Lf+C 2n (_l)j-i

%-body problem, the 2n-ion problem and the n-elec- 1 i<j (4i ’ tron atom for small values of n. This is a new and total- ly unexpected family of classical periodic orbits for piER3,qiER3,i= 1,2 ,..., 2n.Thisisthehamile the above problems general&g the well-known tonian of 2n equal-mass charged particles, the particles Lagrange solution of the equal-mass 3-body problem, 1,3,5,7, . . . . (2n - 1) having charge +l, the particles in which the three bodies are at the vertices of an 2, 4, 6, . . . , 2n charge -1, in some suitable units. The equilateral triangle which rotates rigidly in its plane corresponding equations of motion reduce to about the centroid with an appropriate angular speed [ 11.

These general&d Lagrange solutions were found whilst looking for stable periodic solutions for physic- ally interesting classical hamiltonians such as the n-electron atomic hamiltonian. Such stable periodic orbits would give rise to semi-classical eigenvalues for the corresponding quantum-mechanical hamiltonians [2]. Unlike the 3-body Lagrange solution these new orbits are not planar and are not associated with regu- lar figures such as the equilateral triangle. The sense in which the new solutions generalise the equal-mass 3- body Lagrange solution will be made clear below. Whether or not any of them turns out to be stable re- mains to be seen.

Let us begin by defming the equal-mass 2n-body problem, %-ion problem and n-electron atomic prob- lem. These are the hamiltonian systems below labelled as (H2nI), (Hem2nB) and (HneA).

Example (H2nl). Consider the hamiltonian H(q, p), H: R12”+R,

for i = 1,2, . . . . 2n. We are seeking periodic solutions of these equations

for instance.

Example (HemZnB). In this case the corresponding hamiltonian H : R12n+ R is defined by

H(q, p) = i$ hi” + g -1 i<j ISi-4jI’

PiER3,qrER3,i=1,2 ,..., 2n.Thisisthehamilto- nian of 2n equal-mass gravitating particles. The equa- tions of motion here reduce to

for i = 1,2, . . . . 2n.

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Volume 99A, number 1 PHYSICS LETTERS 14 November 1983

Example (HneA). This is the most physically in- teresting of the hamiltonians which we consider H:l+“+R

H(q, p) = $;pf + 5 ----!_- - i<j 14i - 4j I

The equations of motion for this hamiltonian read

q. = 5 (4i - 4j) nhqi ’ jti jqi - qj 13

-- (X>O), IQi I3

for i = 1,2, . . . . n. These are the equations of motion of n electrons, negatively charged, moving under their mutual repulsions subject to the attraction of an in- finitely massive, positively charged nucleus, with charge nX (usually h = 1).

Before stating a typical instance of our main result we make two further remarks. Firstly the equations of motion above are invariant under the scale trans- formation q(t) + /.~~/~q((ut)@ > 0,4 standing for the collective coordinates (ql, 42, . . . . q2J or (ql, q2, ..,, qn). Secondly the above equations of motion admit solutions of the form

4(t) = (‘II (0, Q(0, ***9 p2n- %I (0) 7

if P E O(3) with P2n = 1 , or

4(0=(41w+71(0, ...J”-141w,

if P E O(3) with P” = 1, respectively. Of course the Lagrange solution of the equal mass 3-body problem is of this kind with P a rotation through 2n/3 about the normal to the plane of the triangle through its centroid.

Theorem. Let P E O(3) be an orthogonal transform- ation of order 2n so that P2n = 1 and let det P = -1. Choose cartesian axes cr, p, 7 so that the cv axis coin- cides with the direction determined by the 1 dimen- sional eigenspace of P for the eigenvalue -1. Then there exists a non-planar periodic orbit of the hamil- tonian system (I-KM) in which, relative to the (Y, 0,~ axes, (11(O) = (0, 1, O), cir(O) = (du,,O, ?o) and 410) =(01,p,r),42(t)=P41(t),43(t)=@41(t),...,42n(t)

= Ps - l q1 (t): if &u, +. satisfy

0.6 -

0 0.2 0.4 0.6 0.6 1

Fig. 1. Graph showing periodic orbits for (H4I) problem in

(~o,+o) plane.

Table 1 (4 o , & values for points marked in fig. 1. The trunaction er- ror is of the order of 2 X lo*.

4/P

0 0 0.676096724 1514 0.504011040 0.505990180 31/8 0.562695312 0.457060546

4/l 0.600918800 0.419778768

912 0.678494140 0.325891601

5/l 0.713338044 0.271732804 11/2 0.733296874 0.235119140

6/l 0.746180549 0.208229641

7/L 0.761665439 0.170739928

8/l 0.770473690 0.145446597 24/l 0.791929568 0.045175998

for coprime integers p and q, where P,, P, and PO are the functions of&o, f. detailed below.

For small n (for each of the three problems above) there appears to be an infiite family of such solutions, one for each sufficiently large (reduced) rational q/p. To illustrate the point, for n = 2, for the hamiltonian system (H4I), the values (ho, qo) giving rise to periodic orbits are plotted in fig. 1. The projections of these periodic orbits are given in fig. 2 for the motion of particle 1 onto the (0,~) and (p, o) planes.

Similar results may well be valid for each n Z 2. Re- sults are valid for each value of n tried so far; for (H2nI),

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Volume 99A, number 1 PHYSICS LETTERS 14 November 1983

(5.Y)

(5.a)

(5,Y)

(B,U)

0 00

90 11/a

Fig. 2. Projection of orbits.

n = 2,3; for (Hem2nB), n = 2,3 ; for (HneA),A = 2,3,4. Let us now explain briefly how to define Pa, P, and

PO. These depend upon &,, and Tu through a positive function f, satisfyfng a second-order non-linear ordin- ary differential equation, designated the f-equation, with boundary conditions (2), (3) depending upon I&, $,-,. (Further details are given in ref. [3] .) Here is our periodic solution fjl (t) = (o, 0, T), o2 = f(r), where r = u2 + r2)lj2 and this equation can be in- verted for rrn Gr G 1, rm being determined by the r,-equation (4) in terms of &,, f0 and f, giving r = = r(a), for 0 ( 01( oM, where j(1) = 0 and fir,)

= a& (see ref. [3] ). For example let us restrict our at- tention to the (H4I) problem. Then thef-equation reads

(+) (=+(4f,;2)1,2 -$-+?)

=(l+$)k( -? l +!?) (4f+ ~2)3/2 + 2 r3

Sf + (4f + 221312 1 ’ (1)

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Volume 99A, number 1 PHYSICS LETTERS

with

f(l) = 0

and

(2)

1

t= .i

[G(r)] - U2 dr .

r(t)

f(1) = 2$/& + 4- l - 2-q

where

(3)

In this example P(o, /I,?) = (4, 7, -P), P E O(3), with detP=-l,p=l.

E=2-1(&;++;)+4-1 -2-‘/2,h=T0.

In this case r,(<l) satisfies

The regularity properties of the tetrahedron 1234, with vertices with position vectors ql, 42, q3, q4, are

summarised by

141(t) - 42(0l= 142(t) - 43011

for our solution &2 = F(a) and

Pa=4 i” [F(cu)]-1’2dol,

0

2E=1/2r +h2/r2 m m - 2/[4flr,) + 2r,$ 1’2 (4)

Setting

r=r(a)

F(a) = 6; - 4 J f’(r) dr (9

r=l [4f(r) + 2rqw ’

(6)

Further

1

where i2 = G(r) and

G(r) = 4flr)

4f(r) +f ‘2(r)

X {2E- l/B-h2/r2 t2/[4f(r)+2r2]1/2). (7)

= kg@) -43w = l44W - 41Wl >

and

191(t) - 43(0l = lqz(f) - 44@) 1 3

“M initially being defined by

QM

&; = 8 s (4a2 t 2r2)-3/2a da .

0

P, = 2 f ‘[G(r)] - 112 dr ,

rm

at each instant of time t. The remarkable fact is that there appears to be an infinite number of such orbits for (H41) and similar results are valid for the problems (I-LM), (Hem2nB) and (HneA), certainly for small values of n, and possibly for each n. Thus, for the clas- sical mechanical model of atomic beryllium, for in- stance, there seems to be an infinite number of clas- sical periodic motions of the orbital electrons, 1, 2,3,4 with above regularity properties. Similar results seem to be valid for all low-lying elements in the periodic table, e.g., helium and lithium. Amazingly enough these results persist for the corresponding Zeeman ef- fect hamiltonians (see ref. [3] ). It would be quite ex- traordinary if these motions did not manifest them- selves in nature.

We are grateful to SERC for partial support through grant GR/C/05168. We would like to thank Dr. F.W. Clarke for providing some early stimulus.

References

[l] L.A. Pars, A treatise on analytical dynamics (Heinemamt, 1968).

14 November 1983

Since Pa = 2P, forces h and f to vanish simultaneously, an alternative expression for oM is mrm)] ‘12.

Finally

PO = h 7 r-2(r)dt, 0

where r(r) satisfies the quadrature

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[2] C. Dewitt-Morette, A. Maheshwari and B. Nelson, Phys. Rep. 50 (1979) 255; A. Voros, Colloq. Int. CNRS 237 (1974) 277; S. Albeverio, Ph. Blanchard and R. Hoegh-Krohn, The trace formula for Schrodinger, preprint, Bielefeld (1980), Materialien XXVII.

[3] I.M. Davies, A. Truman and D. Williams, On classical periodic orbits for n-body problems, in preparation.